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Crystal structures

Tiling by numbers

Nature volume 400, pages 617–618 (1999)Cite this article

There is a long history of overlapping problems in chemistry and geometry. A classic example is the problem of the densest packing of equal spheres. Johannes Kepler conjectured in 1611 that the face-centred cubic lattice ('cubic closest packing') was the solution. In 1883, W. Barlow showed that there is an infinite family of equally dense sphere packings, of which the hexagonal closest packing is the simplest member, and correctly suggested that metallic elements would have such structures. Later, Barlow and Sir William Pope proposed structures based on sphere packings, correctly again, for simple salts. This was exactly what W. H. and W. L. Bragg needed for the first X-ray diffraction solutions of crystal structures, such as those of ZnS and NaCl in 1913.

Nowadays, chemists rely heavily on theoretically derived structures to help determine the structures of new materials, such as zeolites, which are based on low-density atomic packings. For years, the goal of enumerating possible structure types has absorbed the efforts of many crystal chemists, and until now this work has been empirical and intuitive. But for the first time, in a study1 described by Delgado Friedrichs et al. on page 644of this issue, a mathematically rigorous and complete enumeration of the main classes of such structures is achieved. This work is of considerable importance to both chemistry and mathematics.

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Figure 1: Space-filling tiles and networks.

References

  1. Delgado Friedrichs, O., Dress, A. W. M., Huson, D. H., Klinowski, J. & Mackay, A. L. Nature 400, 644–647 (1999).

    Article  ADS  Google Scholar 

  2. Weaire, D. & Phelan, R. Phil. Mag. Letts. 69, 107–110 (1994).

    Google Scholar 

  3. Newsam, J. M., Treacy, M. M. J., Koetsier, W. T. & de Gruyter, C. B. Proc. R. Soc. Lond. A 420, 375–405 (1988).

    Google Scholar 

  4. O'Keeffe, M. Z. Kristallogr. 214, 438–443 (1999).

    Google Scholar 

  5. Grünbaum, B. & Shephard, G. C. Bull. Am. Math. Soc. 3, 951–972 (1980).

    Google Scholar 

  6. O'Keeffe, M. in Foams and Emulsions (eds Sadoc, J. F. & Rivier, N.) 403-421 (Kluwer, Dordrecht, 1999).

  7. Delgado Friedrichs, O. & Huson, D. H. Discr. Comput. Geom. 21, 299–315 (1999).

    Google Scholar 

  8. Conway, J. H. & Sloane, N. J. A. Sphere Packings, Lattices and Groups 3rd edn (Springer, New York, 1999).

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  1. Department of Chemistry, Arizona State University, Tempe, 85287-1604, Arizona, USA

    Michael O'Keeffe

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  1. Michael O'Keeffe
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O'Keeffe, M. Tiling by numbers. Nature 400, 617–618 (1999). https://doi.org/10.1038/23139

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